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Entropy and martingales in Markov chain models

Published online by Cambridge University Press:  14 July 2016

E. Seneta
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Abstract

The concept of entropy in models is discussed with particular reference to the work of P.A.P. Moran. For a vector-valued Markov chain {Xk} whose states are relative-frequency (proportion) tables corresponding to a physical mixing model of a number N of particles over n urns, the definition of entropy may be based on the usual information-theoretic concept applied to the probability distribution given by the expectation . The model is used for a brief probabilistic assessment of the relationship between Boltzmann's Η-Theorem, the Ehrenfest urn model, and Poincaré's considerations on the mixing of liquids and card shuffling, centred on the property of an ultimately uniform distribution of a single particle. It is then generalized to the situation where the total number of particles fluctuates over time, and martingale results are used to establish convergence for .

Keywords

ENTROPYMARKOV CHAINSURN MODELSMARTINGALESFLUCTUATING POPULATION SIZE
Type
Part 6 — Stochastic Processes
Information
Journal of Applied Probability , Volume 19 , Issue A , 1982 , pp. 367 - 381
Copyright
Copyright © 1982 Applied Probability Trust 

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